These are my complete notes for Rotational Motion in Classical Mechanics.
I color-coded my notes according to their meaning - All numbered notes (which I call rules) are red, and include examples and the basis for understanding a topic. Definitions are written in green, and other important information (such as large-scale drawings that are better visualized than explained) was written in blue. All of this information is preserved on this page, with logical flow and breaks. I use ascii line drawings sparingly - If I can convey information or a graph using an image online, I will do so.
All of the knowledge present in these notes are filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Rotational Motion (Classical Mechanics)
Table Of Contents
V. Rotational Motion.
V.I Angular & Tangential Velocity.
# P. Rule 38. The position of an object in rotational motion can be determined in Cartesian and Polar coordinates, going back and forth using trigonometry. When an object is moving along a circle with a constant radius, the object undergoes an angular displacement (with regard to the Polar coordinates (r, θ)):
Δθ = θF - θi
The curved linear distance the object would move when moving along an arc is called the arc length. It is the curved length on the side of the circle outside of the angular displacement Δθ. The equation for arc length is as follows:
s = r × Δθ
Where s is the arc length, r is the radius, and Δθ is the angular displacement. THE UNITS FOR ANGULAR DISPLACEMENT MUST BE IN RADIANS. CONVERT USING (π / 180) IF NEEDS BE.
# P. Rule 39. π = 3.1, give or take. It refers to the ratio of a circle's circumference to its diameter, and as an irrational number, it will go on forever. Since the formula for obtaining pi cancels out any dimensions in length (c/d = (86.9m)/(27.5m) ≈ 3.1415926...), the value of pi is in Radians, which is just a dimensionless placeholder unit. The abbreviation for Radians is 'rad'.
# P. Rule 40. While average linear velocity is conveyed as Vavg = (∆x / ∆t), average angular velocity is written as ωavg = (∆θ / ∆t), which can be represented in Radians per Second, Revolutions per Minute, Degrees per Millisecond, etc.
Radians per Second are most commonly used in Physics, while revolutions per minute is most used in Applicationism and "the real world".
# P. Rule 41. The symbol for angular acceleration is α, which is very convenient considering how similar it looks to a. The average angular acceleration is equal to αavg = (∆ω / ∆t). Therefore, the units for angular acceleration are radians per second squared or revolutions per minute squared.
# P. Rule 42. Just like an object can have uniformly accelerated motion, an object can also have uniformly angularly accelerated motion. This is written as U.α.M., instead of U.A.M., even though the alpha is lowercase. As with linear acceleration, the U.α.M. constants can be used when α is constant. All of the variables from U.A.M. must be converted into their angular form:
ωF, ωi, α, ∆θ, and ∆t.
Now, we can rewrite every U.A.M. equation using these variables:
ωF = ωi + (α × ∆t).
∆θ = (ωi × ∆t) + (1/2 × α × ∆t²)
ωF² = ωi² + (2 × α × ∆θ)
∆θ = (1/2) × (ωF + ωi) × ∆t
ALWAYS use radians in your calculations for U.α.M. equations. Additionally, remember that ωF and ωi are instantaneous angular velocities, as opposed to the average velocity (∆θ / ∆t).
# P. Rule 43. Since angular velocity, angular acceleration, and change in angular position are vectors, they all have direction. However, since clockwise and counterclockwise are observer-dependent, we use the right-hand rule to determine direction.
# Tangential Velocity: The linear velocity of an object moving in a circle. Given by the equation vt = r × ω. Because it is LINEAR, you use m/s as your base units.
# P. Rule 44. The farther along you go on the radius of a circle, the linear distance that point travels when moving through a circle, which is called arc length, will increase. This is in spite of the fact that the angular velocity and angular displacement is the same, regardless of radii length, as each point on the radius covers the same number of degrees in the same amount of time.
The path traveled by each point on the radius will form an internal circle, the point moving around the circumference of this circle. Each point on the radius has a linear velocity when it is moving in the circle, known as tangential velocity, which increases proportionally as the radius and arc length (defined by the formula s = r × Δθ) increase. The tangential velocity is given by the equation vt = r × ω, which, just like the arc length equation, requires radians. The angular velocity in the equation needs to be in radians per second.
See the different internal arc lengths that each radius-length has below:
Tangential Velocity is named for how its velocity vector is tangent to the circumference of the circle, perpendicular to the radius. Tangential Acceleration is the exact same way:
# P. Rule 45. Tangential Acceleration is defined by the following equation:
at = r × α.
Like the equations for arc length and tangential velocity, you must use Radians as your angular quantity. Because it is LINEAR, you use m/s² as your base units.
# P. Rule 46. On the nature of Tangential Velocity (and others) as a vector:
Imagine a situation in which the radius and angular velocity of a point on the circle are both constant - This just means that the circle is rotating at a constant speed.
If this were to be the case, then the linear velocity of the radius, the tangential velocity, would still not be constant, even though tangential velocity is literally equal to the radius times the angular velocity. This is because tangential velocity is a vector with both magnitude and direction. While the magnitude would be constant, the direction of the tangential velocity would be constantly changing (perpendicular to the radius as it makes it revolution around the circle), and so tangential velocity itself would not be constant. Therefore, the radius has neither a tangential acceleration nor an angular acceleration.
V.II Centripetal Velocity.
"Centripetal Acceleration is the linear acceleration that causes the tangential velocity to change direction."
Utterly meaningless on their surface, searching for any possible knowledge in these words is an exercise in futility. However, beyond their nonsensical face-value, an elaboration of these words will reveal their value:
The word 'Centripetal' is derived from "centrum", meaning center, and "petere", meaning to seek. Centripetal acceleration is the acceleration that causes circular motion. Centripetal acceleration is always directed inward to the circle, toward the center. It is a center-seeking linear acceleration. The force inward is what causes circular motion - see Rule 52 for more information.
The equation for Centripetal Acceleration is as follows:
ac = (Vt²) / r
Through the substitution of the substitution of the equation for Tangential velocity, we can simply the equation further:
= ((r × ω)² / r) = ((r² × ω²) / r) = r × ω²
Because it is LINEAR, you use m/s² as your base units.
# P. Rule 48. We can extrapolate the known information about circular centripetal acceleration to apply to any curved path an object may follow, and therefore any object moving on a curved path will experience a centripetal acceleration.
# P. Rule 49. The whole idea of centripetal acceleration and tangential velocity and acceleration and everything else in rotational motion is that all the equations use all the same variables, and you can connect what you have to what you need by repeatedly plugging in numbers into equations. For example, to find centripetal acceleration (equation below), you tangential velocity, which you need the average angular velocity for, and so on. To really instill this into your brain, stare at the equations for rotational motion until you imagine them every time you see a rotating circle.
# P. Rule 50. By the nature of their equations, Tangential velocity requires the object to be in acceleration (either speeding up or slowding down as it goes around the circle), while Centripetal acceleration only needs the object to be going around in the circle at a constant rate.
# P. Rule 51. Newton's Second Law (Rule 70) states that net force equals mass times acceleration, or ΣF = m × a. Since we determined in Rule 48 that every object moving on a curved path will have centripetal acceleration, we can apply centripetal acceleration to Newton's Second Law and end up with the equation for Centripetal Force:
ΣF = m × ac
There are several important rules for comprehending centripetal force:
1) Centripetal Force is not a New Force. Most other forces, such as the force of gravity, tension, friction, and the normal force (Sections V.II - V.IV), are 'independently-defined' forces, meaning they are not dependent on anything else and exist despite of what humans think is possible or impossible. Centripetal force, on the other hand, is composed out of these bedrock forces, whether by a mixture of them or just one. This is because it is the NET force in the inward direction, the net of all the forces acting in the inward direction.
2) Centripetal Force is never in a Free Body Diagram. Because centripetal force is not a new force, it never appears in a free body diagram. In order to determine the centripetal force, you need to draw out your free body diagram and then sum the forces in the inward direction.
3) In is positive, and out is negative. When you sum the forces in the inward direction, the in direction is positive, while the out direction is negative.
# P. Rule 52. Time Constant:
The time constant, τ, is a means of determining the time it takes for an exponential equation to get to a percent of its maximum value, calculated similarly to a half-life. The first time constant is 63.2%, which is a universal constant that must be remembered. It is the result of 1-1/e, exactly. This number comes up with some frequency in physics.
Thus, since the first time constant is 63.2% of the time it takes to reach terminal velocity, then 2τ will be the 86.5%, because it increased by 63.2% of the percent remaining until 100%. 3τ will be 95%, and so on and so on. Every increase in the time constant will make the velocity 63.2% closer to terminal velocity. Thus, it will take 6.91τ for the velocity to be 99.9% of the terminal velocity.
Objects that encounter more air resistance and meet their terminal velocity faster, like coffee filters, will have a small time constant, while an object like a baseball will have a larger time constant.
# P. Rule 53.
# P. Rule 54.
# P. Rule 55.
